Question 1204090
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        If you discard all the verbal husk, then the standard question remains: 


            How long will it take for the deposit of $6500 to get the value of $16,250, 
            if the account is compounded monthly at the annual interest of 7% ?



<pre>
Use the formula for the future value of this compounded account

    16250 = {{{6500*(1+0.07/12)^n}}},


where n is the number of months.


Solve this equation to find n, step by step

    {{{16250/6500}}} = {{{(1+0.07/12)^n}}},

    2.5 = {{{1.005833333^n}}}.


Take logarithm base 10 of both sides

    log(2.5) = n*log(1.005833333)


Find n = {{{log((2.5))/log((1.005833333))}}} = 157.54 months.


Round to the closest {{{highlight(highlight(GREATER))}}} month, in order for the bank was in position
to make the last compounding.


<U>ANSWER</U>.  158 months, or  13 years and 2 months.
</pre>

Solved.



Notice that in this problem, &nbsp;the answer in decimals is &nbsp;NOT &nbsp;ACCEPTABLE.


The answer must be in months, &nbsp;because the month is &nbsp;THE &nbsp;DISCRETE &nbsp;TIME &nbsp;UNIT 
for this problem / (for such problems), &nbsp;due to compounding.



Solving problems for discretely compounded accounts, &nbsp;it is important to understand these things:


(1)  &nbsp;&nbsp;The continuous time &nbsp;IS &nbsp;NOT &nbsp;USED &nbsp;and &nbsp;SHOULD &nbsp;NOT &nbsp;be used in the solution,
     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;since the compounding amount &nbsp;IS &nbsp;NOT &nbsp;a continuous function.


     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The compounded account changes by jumping at the end of each compound period.



(2)  &nbsp;&nbsp;The answer must be in integer number of compounding periods;



(3)  &nbsp;&nbsp;Your goal is not to get precisely the assigned amount - &nbsp;{{{highlight(highlight(NO))}}}.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Your goal is to find the first discrete time, &nbsp;when the compounded account will exceed the assigned amount.



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To see many other similar &nbsp;(and different) &nbsp;solved problems, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/logarithm/Problems-on-discretely-compound-accounts.lesson>Problems on discretely compound accounts</A> 

in this site, &nbsp;and learn the subject from there.


After reading this lesson, &nbsp;you will tackle such problems on your own without asking for help from outside.